Herbrand structure

A structure𝔐 for ℒ is a Herbrand structure if the domain of 𝔐 is the Herbrand universe of ℒ. This fixes the domain of 𝔐, and so each Herbrand structure can be identified with its interpretation, leading to the alternative nomenclature of Herbrand interpretation.

A Herbrand model of a theory T is a Herbrand structure which is a model of T.